ad 1.1

``ad`` Package Documentation
============================

.. contents::

Overview
--------

The ``ad`` package allows you to **easily** and **transparently** perform
**first and second-order automatic differentiation**. Advanced math
involving trigonometric, logarithmic, hyperbolic, etc. functions can also
be evaluated directly using the ``admath`` sub-module.

**All base numeric types are supported** (``int``, ``float``, ``complex``,
etc.). This package is designed so that the underlying numeric types will
interact with each other *as they normally do* when performing any
calculations. Thus, this package acts more like a "wrapper" that simply helps
keep track of derivatives while **maintaining the original functionality** of
the numeric calculations.

From the Wikipedia entry on `Automatic differentiation`_ (AD):

"AD exploits the fact that every computer program, no matter how
complicated, executes a sequence of elementary arithmetic operations
(addition, subtraction, multiplication, division, etc.) and elementary
functions (exp, log, sin, cos, etc.). By applying the chain rule
repeatedly to these operations, derivatives of arbitrary order can be
computed automatically, and accurate to working precision."

Basic examples
--------------

Let's start with the main import that all numbers use to track derivatives::

>>> from ad import adnumber

Creating AD objects (either a scalar or an array is acceptable)::

>>> x = adnumber(2.0)
>>> x
ad(2.0)

>>> y = adnumber([1, 2, 3])
>>> y
[ad(1), ad(2), ad(3)]

>>> z = adnumber(3, tag='z') # tags can help track variables
>>> z
ad(3, z)

Now for some math::

>>> square = x**2
>>> square
ad(4.0)

>>> sum_value = sum(y)
>>> sum_value
ad(6)

>>> w = x*z**2
>>> w
ad(18.0)

Using more advanced math functions like those in the standard math_ and cmath_
modules::

>>> from ad.admath import * # sin, cos, log, exp, sqrt, etc.
>>> sin(1 + x**2)
ad(-0.9589242746631385)

Calculating derivatives (evaluated at the given input values)::

>>> square.d(x) # get the first derivative wrt x
4.0

>>> square.d2(x) # get the second derivative wrt x
2.0

>>> z.d(x) # returns zero if the derivative doesn't exist
0.0

>>> w.d2c(x, z) # second cross-derivatives, order doesn't matter
6.0

>>> w.d2c(z, z) # equivalent to "w.d2(z)"
4.0

>>> w.d() # a dict of all relevant derivatives shown if no input
{ad(2.0): 9.0, ad(3, z): 12.0}

Some convenience functions (useful in optimization)::

>>> w.gradient([x, z]) # show the gradient in the order given
[9.0, 12.0]

>>> w.hessian([x, z])
[[0.0, 6.0], [6.0, 4.0]]

>>> sum_value.gradient(y) # works well with input arrays
[1.0, 1.0, 1.0]

Working with NumPy_ arrays (many functions should work out-of-the-box)::

>>> import numpy as np
>>> arr = np.array([1, 2, 3])
>>> a = adnumber(arr)

>>> a.sum()
ad(6)

>>> a.max()
ad(3)

>>> a.mean()
ad(2.0)

>>> a.var() # array variance
ad(0.6666666666666666)

>>> print sqrt(a) # vectorized operations supported with ad operators
[ad(1.0) ad(1.4142135623730951) ad(1.7320508075688772)]

Main Features
-------------

- **Transparent calculations with derivatives: no or little
modification of existing code** is needed, including when using
the Numpy_ module.

- **Almost all mathematical operations** are supported, including
functions from the standard math_ module (sin, cos, exp, erf,
etc.) and cmath_ module (phase, polar, etc.) with additional convenience
trigonometric, hyperbolic, and logarithmic functions (csc, acoth, ln, etc.).
Comparison operators follow the **same rules as the underlying numeric
types**.

- Nearly all derivative calculations are performed **analytically**
(only the ``gamma`` and ``lgamma`` functions use a high-accuracy
finite difference formula).

- **Real and complex** arithmetic handled seamlessly. Treat objects as you
normally would using the math_ and cmath_ functions, but with their new
``admath`` counterparts.

Installation
------------

You have several easy, convenient options to install the ``ad`` package
(administrative privileges may be required)

1. Download the package files below, unzip to any directory, and run
``python setup.py install`` from the command-line.

2. Simply copy the unzipped ``ad-XYZ`` directory to any other location
that python can find it and rename it ``ad``.

3. If ``setuptools`` is installed, run ``easy_install --upgrade ad``
from the command-line.

4. If ``pip`` is installed, run ``pip --upgrade ad`` from the command-line.

Python 3
--------

To use this package with Python 3.x, you will need to run the ``2to3`` tool at
the command-line using the following syntax while in the unzipped ``ad``
directory::

$ 2to3 -w -f all *.py

This should take care of the main changes required. Then, run
``python3 setup.py install``. If bugs continue to pop up,
please email the author.

Contact
-------

Please send **feature requests, bug reports, or feedback** to
`Abraham Lee`_.

Acknowledgements
----------------

The author expresses his thanks to `Eric O. LEBIGOT (EOL)`_, author of the uncertainties_ package, for providing code insight and inspiration.


.. _NumPy: http://numpy.scipy.org/
.. _math: http://docs.python.org/library/math.html
.. _cmath: http://docs.python.org/library/cmath.html
.. _Automatic differentiation: http://en.wikipedia.org/wiki/Automatic_differentiation
.. _Eric O. LEBIGOT (EOL): http://www.linkedin.com/pub/eric-lebigot/22/293/277
.. _uncertainties: http://pypi.python.org/pypi/uncertainties
.. _Abraham Lee: mailto:tisimst@gmail.com